OFDM based wireless systems require the channel to be determined at the receiver for all subcarriers. Channel estimation for OFDM systems is traditionally approached in the frequency domain, using pilot signals, by estimating the frequency response for a few selected subcarriers and using those observations to interpolate the rest of the subcarriers. Typically, pilot signals or symbols are transmitted over a few subcarriers. Using the received information for those subcarriers, the receiver tries to estimate the channel for the remaining subcarriers. More the number of pilots, more the estimation accuracy. Hence, with this approach, the number of pilots required depends on the coherence bandwidth of the channel; the higher the bandwidth, the lesser the number of pilots.
However, sending pilots involves overhead since bandwidth and power, which could otherwise be used for sending useful data, are expended in sending those pilots. The problem therefore is to minimize the channel estimation overhead while maintaining certain accuracy level for the estimates. The problem becomes even more challenging with multiple antenna systems since there are a lot more channels or channel coefficients to be estimated. Typically, the overhead scales linearly with the number of transmit antenna.
Further, the above approach takes into account only the length of the impulse response and ignores the sparsity of the wireless channel. Wireless channels are typically sparse; the time domain impulse response of the wireless channels typically has a very few nonzero taps, that is, the largest tap delay is usually much greater than the number of nonzero taps.
Channel sparsity is attractive from a system design perspective since it can be exploited to design more efficient channel estimation strategies. It has been shown that a discrete time signal of length M with only T nonzero coefficients can be exactly reconstructed just by observing any 2T samples of its discrete Fourier transform (DFT) if M is prime. Notice that the number of observations required does not depend on the DFT size M. This result has a direct application in high data rate OFDM systems where the number of subcarriers is large. Due to small symbol duration, the length of the discrete time channel response will be large while there may be only very few nonzero coefficients. In such a case, only very few subcarrier pilots are required to estimate the frequency response for all subcarriers.
The proposed optimal signal recovery principle described above is combinatorial in nature. It is essentially an exhaustive search over all possible choices that give rise to the given observations under the sparsity constraint. To avoid this problem, an optimization approach based on L1-norm minimization has been employed for channel recovery in Single Input Single Output (SISO) OFDM systems where the observations are corrupted with noise. The results indicate the scope for potential improvement in wireless channel estimation problems by using the theory of sparse signal recovery where the system model is not necessarily ideal due to noise and M being non-prime in general.
There have been many works on sparse signal recovery, most notably matching pursuit (MP). The MP algorithm iteratively identifies a small subset of the nonzero positions that contribute to most of the energy in the observations. Although the algorithm is suboptimal and greedy in nature, it is efficient in terms of performance and complexity. There are many variants of the algorithm depending on the way the positions are identified. In addition to MP, algorithms based on gradient search have also been proposed for sparse signal recovery.